A small arithmetic hyperbolic three-manifold.

*(English)*Zbl 0621.57006It was shown by Jørgenson and Thurston that there is a minimal element \(v_ 1\) in the set of volumes of complete orientable hyperbolic three- manifolds. Let M be the complete orientable hyperbolic 3-manifold resulting from (5,1) Dehn surgery on the complement of the figure-eight knot K in \(S^ 3\). It is established in this paper that M is arithmetic. On the other hand, the author and Jørgenson have shown that there is an arithmetic manifold M’ with vol M’\(<vol M\).

Reviewer: G.Soifer

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

##### Keywords:

volumes of complete orientable hyperbolic three-manifolds; (5,1) Dehn surgery on the complement of the figure-eight knot; arithmetic manifold
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##### References:

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